Objective of the course – aims of the course unit:
Introduction to the functional analysis, differential operators, survey of the partial differential equations. Boundary and initial conditions. Finite difference methods (FDM).
Finite element methods (FEM). – introduction. Discretization of a region into the finite elements. Approximation of the field from the nodal or edge values.
Forward problem. Setup of equations for nodal and edge values by the Galerkin method.
Application of the Galerkin method to the static and quasistatic fields (Poisson’s and Helmholtz’s equation).
Application of FEM and FDM on the time variable problems (the diffusion and wave equation).
Connection of the field region with the lumped parameter circuit. Coupled problems.
The field optimization problem. Survey of the deterministic methods. The local and global minima.
Unconstrained problems – gradient method, method of the steepest descent, Newton’s methods.
Constrained optimization problems together with FEM.
Inverse problems for the elliptic equations. The Least Square method. Deterministic regularization methods.
A survey on level set methods for inverse problems and optimal design.
A survey on inverse problems in tomography.
A note: Practical examples using the ANSYS and MATLAB environment will be a part of each point of the curriculum.
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Objective of the course – learning outcomes and competences:
To acquire theoretical knowledge as well as practical application of the FEM and FDM together with the ability to program corresponding forward and inverse problems.
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Prerequisites:
Mathematical calculus, Physics, Electromagnetism on the level of MSc.
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Course contents (annotation):
The content of the seminar consists of two related units. The first part deals with the numerical solution of the partial differential equations (PDE), exploiting the Finite Difference method (FDM) and the Finite Element Method. The following PDE are solved by these methods: Laplace’s, Poisson’s, Helmholtz’s, parabolic, and hyperbolic one. The boundary and initial condition as well as the material parameters and source distribution is supposed to be known (forward problem). The connections between the field quantities and the connected circuits as well as the coupled problems are discussed to the end of this part.
The above mentioned FDM and FEM solutions are applied in the second part of the seminar to the evaluation of material parameters of the PDE’s implementing them as a part of the loop of different iterative processes. As the initial values are chosen either some measured data or starting data. The numerical methods utilizing PDE are used for the solution of the optimization problems (finding optimal dimensions or materiel characteristics) and inverse problems (different variants of a tomography known as the Electrical Impedance Tomography, the NMR tomography, the Ultrasound tomography). Each topic is illustrated by practical examples in the ANSYS and MATLAB environment.
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Teaching methods and criteria:
Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.
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Assesment methods and criteria linked to learning outcomes:
Total number of points 100.
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Course curriculum:
Introduction to the functional analysis, differential operators, survey of the partial differential equations. Boundary and initial conditions. Finite difference methods (FDM).
Finite element methods (FEM). – introduction. Discretization of a region into the finite elements. Approximation of the field from the nodal or edge values.
Forward problem. Setup of equations for nodal and edge values by the Galerkin method.
Application of the Galerkin method to the static and quasistatic fields (Poisson’s and Helmholtz’s equation).
Application of FEM and FDM on the time variable problems (the diffusion and wave equation).
Connection of the field region with the lumped parameter circuit. Coupled problems.
The field optimization problem. Survey of the deterministic methods. The local and global minima.
Unconstrained problems – gradient method, method of the steepest descent, Newton’s methods.
Constrained optimization problems together with FEM.
Inverse problems for the elliptic equations. The Least Square method. Deterministic regularization methods.
A survey on level set methods for inverse problems and optimal design.
A survey on inverse problems in tomography.
A note: Practical examples using the ANSYS and MATLAB environment will be a part of each point of the curriculum.
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Specification of controlled education, way of implementation and compensation for absences:
The content and forms of instruction in the evaluated course are specified by the lecturer responsible for the course.
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Recommended reading:
Rektorys Karel: Přehled užité matematiky I, II. Prometheus, 1995
Inverse Problems. IoP Electronic Journals, http://www.iop.org/EJ/journal/IP
http://www.inverse-problems.com/
IEEE Transactions on Magnetics, ročník 1996 a výše
Fotokopie nejnovějších výzkumných zpráv ze serverů zahraničních špičkových pracovišť ke studované problematice budou dodávány průběžně
SIAM Journal on Control and Optimization, ročník 1996 a výše
metoda hladinových množin http://www.math.ucla.edu/applied/cam/index.html
Dědek, L., Dědková J.: Elektromagnetismus. Skripta VUTIUM Brno, 2000
Inverse Problems. IoP Electronic Journals, http://www.iop.org/EJ/journal/IP
Bossavit Alain.: Computational Electromagnetism – Variational formulations, complementarity, edge elements. Academic Press, 1998
Sadiku Mathew: Electromagnetics (second edition), CRC Press, 2001
Chari, M, V. K., Salon S. J.: Numerical Methods in Electromagenetism. Academic Press, 2000
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